Brown and white eggs are dropped by random chickens and snakes and in some cases, Manjini, throughout the game. You can also find them in occassionally in loot. Or on "Chicken Island". (chapter 3-1, middle island)
Gold eggs I have only seen on "Chicken Island" in chapter 3-1. If I've found one elsewhere, I don't remember where I got it.
Rotten eggs drop randomly from chapters 4 on. I got two, from two random Manjini. Like I said, it's completely random, so just hope for the best.
Once you get an egg, put it into your storage until you have the others. Once you have them all in your inventory, the achievement should pop for you. If not, move them from your storage to your active slots. (That's what my friend had to do)
Edit: I found this video on YouTube, so people could get a visual reference. It's not mine, but it shows you where to go to find "Chicken Island", and how to get the brown, white, and golden eggs. All credit for the video goes to: Xplict911.